Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{n^2 - 12n + 32}{n^2 - 10n + 24}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{n^2 - 12n + 32}{n^2 - 10n + 24} = \dfrac{(n - 8)(n - 4)}{(n - 6)(n - 4)} $ Notice that the term $(n - 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(n - 4)$ gives: $a = \dfrac{n - 8}{n - 6}$ Since we divided by $(n - 4)$, $n \neq 4$. $a = \dfrac{n - 8}{n - 6}; \space n \neq 4$